## Taiwanese Journal of Mathematics

### Backward Stability and Divided Invariance of an Attractor for the Delayed Navier-Stokes Equation

#### Abstract

We study backward stability of a pullback attractor especially for a delay equation. We introduce a new concept of a backward attractor, which is defined by a compact, pullback attracting and dividedly invariant family. We then show the equivalence between existence of a backward attractor and backward stability of the pullback attractor, and present some criteria by using the backward limit-set compactness of the system. In the application part, we consider the Navier-Stokes equation with a nonuniform Lipschitz delay term and a backward tempered force. Based on the fact that the delay does not change the backward bounds of the velocity field and external forces, we establish the backward-uniform estimates and obtain a backward attractor, which leads to backward stability of the pullback attractor. Some special cases of variable delay and distributed delay are discussed.

#### Article information

Source
Taiwanese J. Math., Volume 24, Number 3 (2020), 575-601.

Dates
Revised: 4 June 2019
Accepted: 17 June 2019
First available in Project Euclid: 19 May 2020

https://projecteuclid.org/euclid.twjm/1589875220

Digital Object Identifier
doi:10.11650/tjm/190603

#### Citation

Li, Yangrong; Zhang, Qiangheng. Backward Stability and Divided Invariance of an Attractor for the Delayed Navier-Stokes Equation. Taiwanese J. Math. 24 (2020), no. 3, 575--601. doi:10.11650/tjm/190603. https://projecteuclid.org/euclid.twjm/1589875220

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